All Trigonometric Formulas for Government Exams
All Trigonometric Formulas for Government Exams
A comprehensive, colorful guide with formulas, graphs, quadrants, and applications.
Trigonometric Ratios
| Ratio |
Formula |
Description |
| sin θ |
Perpendicular / Hypotenuse |
Opposite side over longest side |
| cos θ |
Base / Hypotenuse |
Adjacent side over longest side |
| tan θ |
Perpendicular / Base |
Opposite over adjacent |
| cot θ |
Base / Perpendicular |
Adjacent over opposite |
| sec θ |
Hypotenuse / Base |
Longest side over adjacent |
| csc θ |
Hypotenuse / Perpendicular |
Longest side over opposite |
Pythagorean Identities
| Identity |
Formula |
| Basic |
sin²θ + cos²θ = 1 |
| Tangent |
1 + tan²θ = sec²θ |
| Cotangent |
1 + cot²θ = csc²θ |
| Rearranged |
sin²θ = 1 - cos²θ |
| Rearranged |
cos²θ = 1 - sin²θ |
Reciprocal and Quotient Identities
| Type |
Formula |
| Reciprocal |
sin θ = 1 / csc θ |
| Reciprocal |
cos θ = 1 / sec θ |
| Reciprocal |
tan θ = 1 / cot θ |
| Quotient |
tan θ = sin θ / cos θ |
| Quotient |
cot θ = cos θ / sin θ |
Angle Sum and Difference Identities
| Function |
Sum Formula |
Difference Formula |
| sin |
sin(α + β) = sin α cos β + cos α sin β |
sin(α - β) = sin α cos β - cos α sin β |
| cos |
cos(α + β) = cos α cos β - sin α sin β |
cos(α - β) = cos α cos β + sin α sin β |
| tan |
tan(α + β) = (tan α + tan β) / (1 - tan α tan β) |
tan(α - β) = (tan α - tan β) / (1 + tan α tan β) |
Double and Half Angle Identities
| Type |
Formula |
| Double (sin 2θ) |
2 sin θ cos θ |
| Double (cos 2θ) |
cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ |
| Double (tan 2θ) |
(2 tan θ) / (1 - tan²θ) |
| Half (sin²(θ/2)) |
(1 - cos θ) / 2 |
| Half (cos²(θ/2)) |
(1 + cos θ) / 2 |
| Half (tan(θ/2)) |
(1 - cos θ) / sin θ = sin θ / (1 + cos θ) |
Product-to-Sum and Sum-to-Product Identities
| Type |
Formula |
| Product-to-Sum |
sin α cos β = (1/2)[sin(α + β) + sin(α - β)] |
| Product-to-Sum |
cos α cos β = (1/2)[cos(α + β) + cos(α - β)] |
| Sum-to-Product |
sin α + sin β = 2 sin((α + β)/2) cos((α - β)/2) |
| Sum-to-Product |
cos α + cos β = 2 cos((α + β)/2) cos((α - β)/2) |
Quadrant Signs
| Quadrant |
sin θ |
cos θ |
tan θ |
cot θ |
sec θ |
csc θ |
| I (0°-90°) |
+ |
+ |
+ |
+ |
+ |
+ |
| II (90°-180°) |
+ |
- |
- |
- |
- |
+ |
| III (180°-270°) |
- |
- |
+ |
+ |
- |
- |
| IV (270°-360°) |
- |
+ |
- |
- |
+ |
- |
Mnemonic: "All Students Take Calculus" (All in I, Sin in II, Tan in III, Cos in IV).
Standard Angle Values
| Angle |
0° |
30° |
45° |
60° |
90° |
| sin θ |
0 |
1/2 |
1/√2 |
√3/2 |
1 |
| cos θ |
1 |
√3/2 |
1/√2 |
1/2 |
0 |
| tan θ |
0 |
1/√3 |
1 |
√3 |
Undefined |
Inverse Trigonometric Identities
| Identity |
Formula |
| Complementary |
sin⁻¹ x + cos⁻¹ x = π/2 |
| Negative |
sin⁻¹ (-x) = -sin⁻¹ x |
| Tangent Sum |
tan⁻¹ x + tan⁻¹ y = tan⁻¹ ((x + y)/(1 - xy)) |
Triangle Formulas
| Type |
Formula |
| Law of Sines |
a/sin A = b/sin B = c/sin C |
| Law of Cosines |
c² = a² + b² - 2ab cos C |
| Area (Sine) |
Area = (1/2)ab sin C |
| Heron’s Formula |
Area = √[s(s-a)(s-b)(s-c)], s = (a+b+c)/2 |
Applications (Heights, Distances, Algebra)
| Application |
Formula/Example |
| Height Calculation |
Height = Distance × tan θ |
| Algebraic Simplification |
Simplify: sin²θ + cos²θ → 1 |
| Equation Solving |
Solve: 2sin θ = 1 → sin θ = 1/2, θ = 30° |
Graphs of Trigonometric Functions
1. Sine Function (sin θ)
Description: Periodic wave, period 360°, amplitude 1, oscillates between -1 and 1. Crosses x-axis at 0°, 180°, 360°.
2. Cosine Function (cos θ)
Description: Similar to sine, shifted left by 90°. Period 360°, amplitude 1, starts at (0,1).
3. Tangent Function (tan θ)
Description: Period 180°, undefined at 90°, 270°. Crosses x-axis at 0°, 180°.
Quadrant Diagram
Description: A circle divided into four quadrants:
- I: (x > 0, y > 0), all positive.
- II: (x < 0, y > 0), sin/csc positive.
- III: (x < 0, y < 0), tan/cot positive.
- IV: (x > 0, y < 0), cos/sec positive.
